The arrow diagram shows a relationship between the sets $P$ and $Q$. Write the relation in what is the domain and range of $\mathrm{ℝ}$.

Let $X=\left\{1, 2, 3, 4\right\}$ and $Y=\left\{2, 4, 6, 8, 10\right\}$ and $\mathrm{ℝ}=\left\{\left(1, 2\right), \left(2, 4\right), \left(3, 6\right), \left(4, 8\right)\right\}$. Show that $\mathrm{ℝ}$ is a function and find its domain, co-domain and range?

A relation '$f$' is defined by $f\left(x\right)=x^2-2$ where, $x\in \left\{-2, -1, 0, 3\right\}$

List the elements of $f$.

A relation '$f$' is defined by $f\left(x\right)=x^2-2$ where, $x\in \left\{-2, -1, 0, 3\right\}$

Is $f$ a function?

If $X=\left\{-5, 1, 3, 4\right\}$ and $Y=\left\{a, b, c\right\}$, then check whether the following relation is function from $X$ to $Y$?

${\mathrm{ℝ}}_1=\left\{\left(-5, a\right), \left(1, a\right), \left(3, b\right)\right\}$

If $X=\left\{-5, 1, 3, 4\right\}$ and $Y=\left\{a, b, c\right\}$, then check whether the following relation is function from $X$ to $Y$?

${\mathrm{ℝ}}_2=\left\{\left(-5, b\right), \left(1, b\right), \left(3, a\right), \left(4, c\right)\right\}$

If $X=\left\{-5, 1, 3, 4\right\}$ and $Y=\left\{a, b, c\right\}$, then check whether the following relation is function from $X$ to $Y$?

${\mathrm{ℝ}}_8=\left\{\left(-5, a\right), \left(1, a\right), \left(3, b\right), \left(4, c\right), \left(1, b\right)\right\}$

Given $f\left(x\right)=2x-x^2,$ find $f\left(1\right)$

Given $f\left(x\right)=2x-x^2,$ find $f\left(x+1\right)$